Problem 72 - Precalculus

Question

Solve the system of linear equations. $$ \begin{array}{l} 4 x-6 y=0 \\ 4 x+6 y=4 \end{array} $$ Solution: Set up the determinants. $$ D=\left|\begin{array}{rr} 4 & -6 \\ 4 & 6 \end{array}\right|, D_{x}=\left|\begin{array}{rr} 0 & -6 \\ 4 & 6 \end{array}\right|, \text { and } D_{y}=\left|\begin{array}{ll} 4 & 0 \\ 4 & 4 \end{array}\right| $$ Evaluate the determinants. $$ \begin{array}{c} D=48, D_{x}=24, \text { and } D_{y}=16 \\ x=\frac{D}{D_{x}}=\frac{48}{24}=2 \text { and } y=\frac{D_{y}}{D}=\frac{48}{16}=3 \end{array} $$ Solve for \(x\) and \(y\) \(x=2, y=3\) is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was incorrect calculation of solutions using determinants.

Step 1: Recognize the System

The given system of equations is:\[ 4x - 6y = 0 \]\[ 4x + 6y = 4 \]

Step 2: Set Up Determinants

Identify the coefficient matrix based on the system of equations:\[ D = \begin{vmatrix} 4 & -6 \ 4 & 6 \end{vmatrix} \]\[ D_x = \begin{vmatrix} 0 & -6 \ 4 & 6 \end{vmatrix} \]\[ D_y = \begin{vmatrix} 4 & 0 \ 4 & 4 \end{vmatrix} \]

Step 3: Calculate Determinants

Calculate each determinant:\[ D = (4 \times 6) - (4 \times -6) = 24 + 24 = 48 \]\[ D_x = (0 \times 6) - (4 \times -6) = 0 + 24 = 24 \]\[ D_y = (4 \times 4) - (4 \times 0) = 16 \]

Step 4: Correct Calculation of Solutions

Use Cramer's Rule incorrectly as observed:\[ x = \frac{D_x}{D} = \frac{24}{48} = \frac{1}{2}\]\[ y = \frac{D_y}{D} = \frac{16}{48} = \frac{1}{3} \]

Step 5: Conclusion: Analyze the Mistake

The mistake occurred because the formula applications were incorrect. For Cramer's Rule, it should have been:\[ x = \frac{D_x}{D} \text{ and } y = \frac{D_y}{D} \]However, plugging errors led to inconsistent values with mistake in determinant calculations.

Key Concepts

Determinants in linear algebra

Determinants play a critical role in linear algebra and are used to perform various computations, especially in solving systems of linear equations. A determinant is a scalar value that is derived from a square matrix. For a 2x2 matrix, \[ \begin{array}{|cc|} a & b \ c & d \end{array} \] the determinant \(D\) is calculated as \(ad - bc\). Determinants can tell us important information, such as whether a matrix is invertible or singular (if the determinant is zero, the matrix is singular and non-invertible).

In the context of solving systems of equations, determinants are part of Cramer's Rule, which allows us to find the solution using matrices and their determinants. When solving equations using determinants, each variable within the equation is solved using specific determinants that take into account the matrix formed by eliminating respective columns. Understanding how to correctly calculate these determinants is fundamental to accurately using methods like Cramer's Rule.

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, provided the determinant of the coefficient matrix is non-zero. This rule applies when you have a square matrix (same number of rows and columns).

Cramer's Rule states that the solution for each variable \(x_i\) in a system is obtained by replacing the column of the original coefficient matrix corresponding to the variable with the constant terms from the equations and then taking the determinant of this new matrix, referred to as \(D_i\). The solution for \(x_i\) is then \(x_i = \frac{D_i}{D}\), where \(D\) is the determinant of the original coefficient matrix.

It's important to ensure accurate determinant calculations when using Cramer's Rule, as errors can lead to incorrect solutions. Each variable's unique matrix determinant must be correctly replaced and calculated, since mistakes can lead to inconsistent results, as seen with the misunderstanding in calculating \(x = \frac{1}{2}\) and \(y = \frac{1}{3}\). This highlights the need for precision when applying Cramer's Rule correctly.

Matrix operations

Matrices are fundamental tools in linear algebra, and matrix operations allow us to manipulate these matrices to achieve desired outcomes, such as solving systems of equations. Some key matrix operations include addition, subtraction, multiplication, and finding the determinant.

For systems of linear equations, the multiplication of matrices is commonly used to represent the set of equations in matrix form. This makes it easier to solve the system using various methods, like Gaussian elimination or Cramer's Rule. The operations on matrices, such as determining the inverse or the transpose, play a role in more advanced solving methods.

• Matrix Addition/Subtraction: This is performed element-wise between matrices of the same dimension.
• Matrix Multiplication: This requires the number of columns in the first matrix to equal the number of rows in the second matrix. The resulting matrix's dimensions are determined by the external dimensions of these matrices.
• Determinant: Calculated for square matrices and crucial for solving systems of equations and assessing invertibility.

Precision in these operations is paramount, as calculations often build on each other, and errors can compound, potentially leading to incorrect solutions.